What is the limit of the function f(x) = sin(x)/x as x approaches zero?

To find the limit of the function ( f(x) = \frac{\sin(x)}{x} ) as ( x ) approaches zero, we can utilize the well-known limit property in calculus. This limit can be analyzed through various methods, including using L'Hôpital's Rule, the Squeeze Theorem, or recognizing the Taylor series expansion of ( \sin(x) ).

When we approach this limit directly by substituting ( x = 0 ), we encounter an indeterminate form ( \frac{0}{0} ). However, from calculus, we know that as ( x ) gets very close to zero, the value of ( \sin(x) ) behaves similarly to ( x ). More specifically, the Taylor series expansion for ( \sin(x) ) around zero is:

[

\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots

]

When substituting ( \sin(x) ) into the limit expression we have:

[

\lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac